DK. W. J. MACQTTORN EANKINE ON PLANE WATEE-LINES. 
383 
trajectories to a set of oogenous neo'ids. The function q, as is well known, must satisfy 
the equation 
dq db , dq db q 
dx dx dy dy 
Referring to equation (19) of article 7 for the value of b , it is easily seen that this con- 
dition is fulfilled by the following function, 
*=•+£■** **■ | < 43 ) 
which has also the following properties, 
dq db u dq db v d*q dfq „ 
— = - — — - — — - : 
dx 
dy c ’ dy dx c’ dx 2 1 dy 2 v ' *'’*** 
Every orthogonal trajectory has a straight asymptote parallel to the axis of y, and 
expressed by the equation x=q. 
The perpendicular distance between two consecutive orthogonal trajectories, like 
that between two consecutive water-lines, is inversely proportional to the velocity of 
gliding; hence, if a complete set of orthogonal trajectories were drawn on fig. 1, they 
would divide it into a network of small rectangles, the dimensions and area of any one 
of which would be expressed as follows : — 
cdb cdq __ cHbdq 
a / m 2 + v 2 VvP + v 2 
(45) 
For a series of cyclogen ous neoids, the equation of the orthogonal trajectories takes the 
following form, 
( 45a ) 
(20.) Disturbances of Pressure and Level . — Let h denote the head at a given particle 
of liquid, being the sum of its elevation above a fixed level and of its pressure, expressed 
in units of height of the liquid itself. In a mass of liquid which is at rest, the head 
has a uniform value for every particle of the mass ; let that value be denoted by h 0 . 
Then when the mass of liquid is in the state of motion produced by the passage of a 
solid through it, the head at each particle, according to well-known principles, under- 
goes the change expressed by the following equation, 
d 2 — 7/ 2 — 
h — L— 
2y 
(46) 
being the height due to the difference between the squares of the speed of the solid body 
and of the speed of gliding ; and in an open mass of water with a vessel floating in it, that 
change will take place by alterations in the level of surfaces of equal pressure. The tra- 
jectory of slowest gliding, LN (Plate VIII. fig. 1), will mark the summit of a swell thus 
produced, and so also will the axis of y between O andP ; while the trajectory of swiftest 
gliding OP, and the axis of y beyond P, will mark the bottom of a hollow. These are the 
principal vertical disturbances, which, throughout this investigation, have been assumed 
MDCCCLXIV. 3 F 
